On the semisimplicity of low-dimensional representations of semisimple groups in characteristic \(p\)
type: Item
instance of: scholarly article
title: On the semisimplicity of low-dimensional representations of semisimple groups in characteristic \(p\)
zbMATH Open document ID: 0834.20045
author: Michael Larsen
published in: Journal of Algebra
publication date: 1996-03-20T00:00:00Z
review text: The main result of the paper is that any module of dimension at most \(\sqrt{p/3}\) for a semisimple algebraic group is semisimple if the characteristic of the ground field is \(p\). In the process of proving this result, the author also proves that the tensor product of two semisimple modules is semisimple provided the dimension of the tensor product module is at most \(p/f-h\) with \(f\) being the index of the root lattice in the weight lattice and \(h\) being the Coxeter number. The situation for finite groups of Lie type is also considered. It should be pointed out that Jantzen has proved that the modules with dimensions at most \(p\) are semisimple and \textit{J.-P. Serre} [Invent. Math. 116, 513-530 (1994; Zbl 0816.20014)] has proved a much stronger result for the tensor product of semisimple modules to be semisimple. The readers should be warned that the definition of alcoves in the paper and the alcoves later referred are not consistent.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 763942
zbMATH Keywords: semisimple algebraic groups
zbMATH Keywords: root lattices
zbMATH Keywords: weight lattices
zbMATH Keywords: Coxeter numbers
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: tensor products of semisimple modules
zbMATH Keywords: alcoves
reviewed by: Zongzhu Lin
MaRDI profile type: Publication
full work available at URL: jabr.1995.1085
OpenAlex ID: W2053821684
DOI: 10.1006/JABR.1995.1085
description: scientific article; zbMATH DE number 763942